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How to remove duplicate points based on other attributes

I have a point dataset of hundreds of thousands of points (zip+4s), many of which share the same value for one attribute (zip+4), but have different values for other attributes (ex: address). In other words, each zip+4 is represented by multiple points at different addresses. I need a dataset that has only one record for each zip+4. In the past, I have simply dissolved the entire dataset by zip+4 and ended up with multi-point features. This, however is starting to cause problems down the line, because I need the location of each zip+4 to be unique - and if a single zip+4 crosses into multiple cities or counties, it gives me issues.

I have gotten as far as dissolving them into smaller multi-point features by city/county, so that each unique combination of zip+4, city, and county is represented by it's own multi-point feature (and I retained a count attribute of how many points make up each multi-point feature). Now how do I get rid of all but the one record for each zip+4 that has the most points in it?